Curved Spaces. From Classical Geometries to Elementary Differential Geometry
An introduction to the differential geometry of curves and surfaces is usually contained in the list of basic courses for students of mathematics at undergraduate level. There are excellent books already available that cover the topic well. This book offers, however, a different approach to these standard topics. It starts with the principal examples of classical geometries - Euclidean, spherical, toric and hyperbolic. The author discusses their basic properties, explains their symmetry group and then uses these explicit examples of geometries together with explicit computations in these basic models to introduce many important notions (isometries, curves and their lengths, special forms of the Gauss-Bonnet theorem, triangulations and Riemann surfaces of higher genus). He then introduces Riemannian metrics on open subsets of R2, defines regularly embedded surfaces in space and discusses geodesics on such surfaces. The book culminates with a definition of abstract surfaces and the Gauss-Bonnet theorem. The book is written in a nice and precise style and explicit computations and proofs make the book easy to understand. A detailed and explicit discussion of the main examples of classical geometries contributes well to a better understanding of later generalisations. A list of examples at the end of each chapter helps as well. It is a very good addition to the literature on the topic and can be very useful for teachers preparing their courses as well as for students.